2002

# A series expansion of fractional Brownian motion

## Publication

### Publication

Let $B$ be a fractional Brownian motion with Hurst index $H in (0,1)$. Denote by $x_1 < x_2 < cdots$ the positive, real zeros of the Bessel function $J_{-H$ of the first kind of order $-H$, and let $y_1 < y_2 < cdots$ be the positive zeros of $J_{1-H$. We prove the series representation begin{equation* B_t = sum_{n=1^infty frac{sin x_n t{x_n, X_n + sum_{n=1^infty frac{1-cos y_n t{y_n, Y_n, end{equation* where $X_1, X_2, ldots$ and $Y_1, Y_2, ldots$ are independent, Gaussian random variables with mean zero and $Var X_n = 2c_H^2x_n^{-2HJ^{-2_{1-H(x_n)$, $Var Y_n = 2c_H^2y_n^{-2HJ^{-2_{-H(y_n)$, where the constant $c_H^2$ is defined by $c_H^2 = pi^{-1Gamma(1+2H)sin pi H$. With probability $1$, both random series converge absolutely and uniformly in $t in [0,1]$.

Additional Metadata | |
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Gaussian processes (msc 60G15), Self-similar processes (msc 60G18), Bessel and Airy functions, cylinder functions, ${}_0F_1$ (msc 33C10) | |

Logistics (theme 3), Energy (theme 4) | |

CWI | |

CWI. Probability, Networks and Algorithms [PNA] | |

Organisation | Stochastics |

Dzhaparidze, K.O, & van Zanten, J.H. (2002).
A series expansion of fractional Brownian motion. CWI. Probability, Networks and Algorithms [PNA]. CWI. |